Let us now consider that there are two countries, i=1,2. Countries are solely characterized by a difference in the return to education in the production function. We assume that the following inequality ε
^{1}>ε
^{2} holds for the rest of the theoretical analysis. There are no other differences between countries. In country 2 the productivity of education is higher than in country 1, since \(\varepsilon ^{i}\in \left [0,1\right ]\).
5.1 Incentives for permanent international migration
Suppose that labor is permitted to migrate internationally. Let us assume that only the young can permanently migrate. Migrants spend their education time, their working time, as well as their leisure or their retirement time over the three periods in the immigration country. The borders between countries are supposed to be opened at time t−1=0.
Proposition 1.
As long as \(\log \left [\frac {\varepsilon ^{2}{e_{1}^{1}}}{\varepsilon ^{1}{e_{1}^{2}}}\right ]<\beta \log \left [ \frac {{Q_{2}^{2}}}{{Q_{2}^{1}}} \right ]\), international migration is unilateral. Rational individuals born in country i have an incentive for permanent migration in country j, where i≠j.
Proposition 1 shows that permanent migration occurs if the utility of the ratio of the marginal returns to education of country 1 over country 2 is less than the discounted utility of the ratio of the production of country 2 over country 1^{2}.
Proof.
Rational individuals born in country 1 have an incentive for permanent migration in country 2 if their indirect utility evaluated at the steadystate price system of country 2 over their lifecycle is higher than their indirect utility evaluated at the steadystate prices of country 1. The condition is:
$$\log {c_{1}^{1}} + \gamma \log \left(1{\ell^{1}_{1}}\right)+\beta \log {d_{2}^{1}} <\log {c_{1}^{2}}+ \gamma \log \left(1{\ell^{2}_{1}}\right)+\beta \log {d_{2}^{2}}. $$
Note that we know from the previous sections that the labor supply is an increasing function of the return to education, see (33) in Appendix Appendix A, so that we have the following relationship
$$\gamma \log \left(1{\ell^{1}_{1}}\right) <\gamma \log \left(1{\ell^{2}_{1}}\right). $$
We now prove that
$$\begin{aligned} \log {c_{1}^{1}} +\beta \log {d_{2}^{1}}& <\log {c_{1}^{2}}+\beta \log {d_{2}^{2}}, \\ \log \left[\frac{{c_{1}^{1}}}{{c_{1}^{2}}}\right]& <\beta \log \left[ \frac{{d_{2}^{2}}}{{d_{2}^{1}}} \right]. \end{aligned} $$
Using relation (32) in Appendix Appendix A
$${c_{1}^{i}}=\frac{1\sigma}{\beta (1\sigma \nu)}{K_{2}^{i}}, $$
and using (13), we have
$$ {K_{2}^{i}}=\frac{a(1\sigma\nu)}{\sigma \varepsilon^{i}}{e^{i}_{1}}, $$
which we put into the previous consumption relation. We now have
$${c_{1}^{i}}=\left[\frac{1\sigma}{\sigma}\right]\left[\frac{a}{ \beta\varepsilon^{i}}\right]e^{i}_{1}. $$
Replace these expressions into the condition relative to the incentives for permanent migration
$$\log \left[\frac{\varepsilon^{2}{e_{1}^{1}}}{\varepsilon^{1}{e_{1}^{2}}}\right]<\beta \log \left[ \frac{{d_{2}^{2}}}{{d_{2}^{1}}} \right]. $$
Using relation (31) in Appendix Appendix A, we have
$$\log \left[\frac{\varepsilon^{2}{e_{1}^{1}}}{\varepsilon^{1}{e_{1}^{2}}}\right]<\beta \log \left[ \frac{{Q_{2}^{2}}}{{Q_{2}^{1}}} \right]. $$
As long as ε
^{1}
e
^{2}>ε
^{2}
e
^{1}, the left hand side is always negative so that the condition is satisfied, considering that in the right hand side, the ratio of production is greater than one^{3}. □
5.2 Dynamics with permanent international migration
Subsection 5.2 is devoted to the study of the dynamics of capital in country 2 and country 1. Without loss of generality, we consider that incentives for migration are directed from country 1 to country 2. In this model, only the young are permitted to permanently migrate from country 1 to country 2. In steadystate equilibrium, period t−1=0, borders are open. A fraction m
^{i} of the young is allowed to migrate. As it will be shown, m
^{i} may be positive or negative depending on the direction of the incentives for international migration. Consequently, according to the previous Subsection 5.1, m
^{1}<0 characterizes the fact that individuals emigrate from country 1, while m
^{2}>0 characterizes the fact that individuals immigrate in country 2.
Since after migration individuals are identical in each country — they train in the home country if they do not migrate, or they train abroad if they migrate — in a given period t≥2, the population in country 2 is \({L_{t}^{2}}={\ell _{t}^{2}}+m^{2} {\ell _{t}^{1}}=\left (1+m^{2}\right){\ell _{t}^{2}}\), while the population in country 1 is \({L_{t}^{1}}=\left (1m^{1}\right) {\ell _{t}^{1}}\). Consequently, in each country, efficient labor is defined as \({L_{t}^{2}}e_{t1}^{\varepsilon ^{2}}=\left (1+m^{2}\right){\ell _{t}^{2}}e_{t1}^{\varepsilon ^{2}}\) and \({L_{t}^{1}}e_{t1}^{\varepsilon ^{1}}=\left (1m^{1}\right) {\ell _{t}^{1}}e_{t1}^{\varepsilon ^{1}}\). The production function of country 2 is
$$\begin{aligned} &{Q_{t}^{2}}=\left({K_{t}^{2}}\right)^{1\sigma\nu}\left(1+m^{2}\right)^{\sigma}\left({\ell_{t}^{2}} e_{t1}^{\varepsilon^{2}}\right)^{\sigma}\left(1+m^{2}\right)^{\nu}\theta_{t}^{\nu}\\ \ \ &\!\!\Longleftrightarrow \ \ {Q_{t}^{2}}=\left(1+m^{2}\right)^{\sigma+\nu}\left({K_{t}^{2}}\right)^{1\sigma\nu}\left({l_{t}^{2}}e_{t1}^{\varepsilon^{2}}\right)^{\sigma}\theta_{t}^{\nu}. \end{aligned} $$
The production function of country 1 is
$$\begin{aligned} &{Q_{t}^{1}}=\left({K_{t}^{1}}\right)^{1\sigma\nu}\left(1m^{1}\right)^{\sigma}\left({\ell_{t}^{1}} e_{t1}^{\varepsilon^{1}}\right)^{\sigma}\left(1m^{1}\right)^{\nu}\theta_{t}^{\nu} \\ \ &\!\!\Longleftrightarrow \ \ {Q_{t}^{1}}=\left(1m^{1}\right)^{\sigma+\nu}\left({K_{t}^{1}}\right)^{1\sigma\nu}\left({l_{t}^{1}}e_{t1}^{\varepsilon^{1}}\right)^{\sigma}\theta_{t}^{\nu}. \end{aligned} $$
Note that there are no indexes on the old efficient labor since whatever the country, old efficient labor supply is the same. A rational firm in country i=1,2 maximizes its profit
$$ \begin{aligned} &\max_{{K_{t}^{2}},{\ell_{t}^{2}},{\theta_{t}^{2}}} \left(1\,+\,m^{2}\right)^{\sigma+\nu}\left({K_{t}^{2}}\right)^{1\sigma\nu}\left({\ell_{t}^{2}}e_{t1}^{\varepsilon^{2}}\right)^{\sigma}\theta_{t}^{\nu} \,\,{w_{t}^{2}}\left(1\,+\,m^{2}\right){\ell_{t}^{2}}e_{t1}^{\varepsilon^{2}}\,\,{p_{t}^{2}} \left(1\,+\,m^{2}\right){\theta_{t}^{2}}\,\,{R_{t}^{2}}{K_{t}^{2}}, \\ &\max_{{K_{t}^{1}},{\ell_{t}^{1}},{\theta_{t}^{1}}} \left(1\,\,m^{1}\right)^{\sigma+\nu}\left({K_{t}^{1}}\right)^{1\sigma\nu} \left({\ell_{t}^{1}}e_{t1}^{\varepsilon^{1}}\right)^{\sigma}\theta_{t}^{\nu} \,\,{w_{t}^{1}}\left(1\,\,m^{1}\right){\ell_{t}^{1}}e_{t1}^{\varepsilon^{1}}\,\, {p_{t}^{1}}\left(1\,\,m^{1}\right){\theta_{t}^{1}}\,\,{R_{t}^{1}}{K_{t}^{1}}. \end{aligned} $$
The first order condition for country i=1,2 where m
^{i} is positive for i=2 or negative for i=1
$$\begin{array}{*{20}l} (1\sigma\nu)\frac{{Q_{t}^{i}}}{1+m^{i}}&={R_{t}^{i}}\frac{{K_{t}^{i}}}{1+m^{i}}, \end{array} $$
((16))
$$\begin{array}{*{20}l} \sigma \frac{{Q_{t}^{i}}}{1+m^{i}}&={w_{t}^{i}}{\ell_{t}^{i}}e_{t1}^{\varepsilon^{i}}, \end{array} $$
((17))
$$\begin{array}{*{20}l} \nu \frac{{Q_{t}^{i}}}{1+m^{i}}&={p_{t}^{i}}{\theta_{t}^{i}}. \end{array} $$
((18))
Note that the following relations are unchanged compared with autarkic equilibrium, but now, due to migration flows, the population can no longer be normalized to unity as was the case in autarky. The dynamics of country 2 and country 1 are
$$\begin{aligned} K_{t+1}^{2}&=\left(1+m^{2}\right){s_{t}^{2}}, \\ K_{t+1}^{1}&=\left(1m^{1}\right){s_{t}^{1}}. \end{aligned} $$
Consequently, considering that m
^{2}>0 and m
^{1}<0, the individual’s first and second period budget constraints are modified as follows
$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{rcl} {ae}_{t1}^{i}&=&E_{t1}^{i},\\ {c_{t}^{i}}+\frac{k_{t+1}^{i}}{1+m^{i}}+{R_{t}^{i}}a e_{t1}^{i}&=&{w_{t}^{i}} {\ell_{t}^{i}} \left(e_{t1}^{i}\right)^{\varepsilon^{i}}, \\ d_{t+1}^{i}&=&R_{t+1}^{i}\frac{k_{t+1}^{i}}{1+m^{i}}+p_{t+1}^{i}\theta_{t+1}^{i}. \end{array} \right. \end{array} $$
Using exactly the same procedure as in autarky, we obtain the new expressions of the consumption of the old
$$d_{t+1}^{i}=(1\sigma)\frac{Q_{t+1}^{i}}{1+m^{i}}, $$
the consumption of the young
$$c_{t}^{i}=\left[\frac{1\sigma}{\beta(1\sigma\nu)}\right]\frac{K_{t+1}^{i}}{1+m^{i}}, $$
the adult and old labor are unchanged, and finally
$$e_{t1}^{i}=\frac{\varepsilon^{i}\sigma }{(a(1\sigma\nu))}\frac{{K_{t}^{i}}}{1+m^{i}}. $$
Property 2.
The old efficient labor supply is independent of the returns to education, ε
^{i}, i.e., there is labor market integration of migrants when old.
Using the second period budget constraint, we can easily compute the steadystate capital per worker in each country.
$$\begin{array}{@{}rcl@{}} \hat K^{2}\,=\, \left[\frac{\beta(1\sigma\nu)\sigma\left(1\,\,\varepsilon^{i}\right)\left(1+m^{2}\right)^{\nu+\sigma\left(1\varepsilon^{i}\right)}} {(1\sigma)+\beta(1\sigma\nu)}\left[\frac{\varepsilon^{i}\sigma}{a(1\sigma\nu)}\right]^{\varepsilon^{i}\sigma}{\ell^{i}}^{\sigma}{\theta^{i}}^{\nu} \right]^{\frac{1}{\nu+\sigma\left(1\varepsilon^{i}\right)}},\\ \end{array} $$
((19))
$$\begin{array}{@{}rcl@{}} \hat K^{1}\,=\,\left[\frac{\beta(1\sigma\nu)\sigma\left(1\varepsilon^{i}\right)\left(1m^{1}\right)^{\nu+\sigma\left(1\varepsilon^{i}\right)}} {(1\sigma)+\beta(1\sigma\nu)}\left[\frac{\varepsilon^{i}\sigma}{a(1\sigma\nu)}\right]^{\varepsilon^{i}\sigma}{\ell^{i}}^{\sigma}{\theta^{i}}^{\nu} \right]^{\frac{1}{\nu+\sigma\left(1\varepsilon^{i}\right)}}.\\ \end{array} $$
((20))
Since both postmigration economies converge to a market steadystate equilibrium, we now investigate by which migration policy the social planner can guide the economy towards a firstbest static welfare optimum. In standard overlapping generations models, this is designated as the Golden Rule, and the government would calculate a tax system that leads the static capital per capita to maximize total consumption in that static state. Our problem is not exactly the same for two reasons. The first reason is that there is no tax system in our economy, and the second reason is that our problem is multidimensional. Since there is no tax system, the government uses the migration rate as a policy instrument in order to choose the static welfare maximizing level of education, adult and old labor, consumption, as well as the capital per worker ratio. Consequently, we must reformulate the social planner’s problem, and this is the objective of the next subsection.
5.3 The static welfare optimum with permanent international migration
We define the static welfare optimum of the economy and examine how it can be reached. It is defined as the stationary state that a social planner would select to maximize welfare under the feasibility constraint. The welfare criterion a collectivity must choose in order to rank all possible steady states has usually been described — following Samuelson (1958) — as the one that maximizes aggregate consumption. In standard models, this is called the Golden Rule, and the government would calculate the static capital per capita that achieves this. Our problem is slightly different in the sense that now the social planner of each country i=1,2 maximizes the static welfare, and by doing this, he chooses the optimal levels of education \({e_{w}^{i}}\) (where the subscript w captures the welfare maximizing solution of each variable), adult labor \({\ell ^{i}_{w}}\) and old labor \({\theta ^{i}_{w}}\), adult and old consumptions \({c^{i}_{w}}\) and \({d^{i}_{w}}\), as well as the capital per worker \({k^{i}_{w}}\). He uses the level of migration m
^{i} as an instrument to guide the economy toward the static welfare optimum, taking into account the macroeconomic equilibrium constraint of his country.
In the integrated world economy, the benevolent social planner in each country i=1,2 solves the following problem
$$\max_{{K^{i}_{w}},{\ell^{i}_{w}},{\theta^{i}_{w}},{e^{i}_{w}},{c^{i}_{w}},{d^{i}_{w}}} \log\left[c_{w}^{i}\right]+\gamma \log\left(1{\ell^{i}_{w}}\right)+\beta \log \left[{d_{w}^{i}}\right] + \beta\gamma \log\left(1{\theta_{w}^{i}}\right), $$
subject to the macroeconomic equilibrium constraint
$$a{e_{w}^{i}}+ {c_{w}^{i}} +{d_{w}^{i}} + {K_{w}^{i}}={{K_{w}^{i}}}^{1\sigma\nu}\left({\ell_{w}^{i}}e_{w}^{\varepsilon^{i}}\right)^{\sigma} \theta_{w}^{\nu}. $$
In each country i=1,2 the first order condition is
$$\begin{array}{*{20}l} (1\sigma\nu){Q_{w}^{i}}& = {K_{w}^{i}}, \end{array} $$
((21))
$$\begin{array}{*{20}l} \frac{\sigma {Q_{w}^{i}}}{{c_{w}^{i}}{\ell_{w}^{i}}} &= \frac{\gamma}{1{\ell_{w}^{i}}}, \end{array} $$
((22))
$$\begin{array}{*{20}l} a{e_{w}^{i}}&=\varepsilon^{i}\sigma {Q_{w}^{i}}, \end{array} $$
((23))
$$\begin{array}{*{20}l} \frac{\nu {Q_{w}^{i}}}{{c_{w}^{i}}\theta^{i}}&=\frac{\beta\gamma} {1{\theta_{w}^{i}}}, \end{array} $$
((24))
$$\begin{array}{*{20}l} {d_{w}^{i}} &=\beta {c_{w}^{i}}. \end{array} $$
((25))
The post migration macroeconomic constraint of the country 2 is as follows
$${c_{w}^{i}}={Q_{w}^{i}}a{e_{w}^{i}}{d_{w}^{i}}{K_{w}^{i}}. $$
Using (21), (22), and (25) and isolating \(\frac {Q^{i}}{c^{i}}\) gives
$$\begin{array}{@{}rcl@{}} \frac{{Q_{w}^{i}}}{{c_{w}^{i}}}=\frac{(1+\beta)}{\nu +\left(1\varepsilon^{i}\right)\sigma}. \end{array} $$
((26))
Putting the last expression into (22) and isolating \({\ell _{w}^{i}}\) gives the optimal adult labor \({\ell _{w}^{i}}\) in each country i=1,2
$$\begin{array}{@{}rcl@{}} {\ell_{w}^{i}}=\frac{\sigma(1+\beta)}{\sigma(1+\beta)+\gamma\left[{\nu}+{\sigma}\left(1\varepsilon^{i}\right)\right]}. \end{array} $$
((27))
Also, putting (26) into (24) and isolating \({\theta _{w}^{i}}\) gives the optimal old labor in each country i=1,2
$$\begin{array}{@{}rcl@{}} {\theta_{w}^{i}}=\frac{{\nu}(1+\beta)}{{\beta}{\gamma}\left[{\nu}+{\sigma}\left(1\varepsilon^{i}\right)\right]+{\nu}(1+\beta)}. \end{array} $$
((28))
Using (21) in (23) and isolating e, we find the expression of the chosen level in education in country i
$$\begin{array}{@{}rcl@{}} {e^{i}_{w}}=\frac{\varepsilon^{i}\sigma {K_{w}^{i}}}{(1\sigma\nu)a}. \end{array} $$
((29))
From relation (21) we deduce the optimal capital per worker that maximizes the welfare in each country
$$\begin{array}{@{}rcl@{}} {K^{i}_{w}}=\left[(1\sigma\nu)\left(\frac{\varepsilon^{i}\sigma}{(1\sigma\nu)a}\right)^{\varepsilon^{i}\sigma}\ell^{\sigma}_{w}\theta^{\nu}_{w}\right]^{\frac{1}{\nu+\sigma\left(1\varepsilon^{i}\right)}}. \end{array} $$
((30))
Proposition 2.

1.
If the return to education is lower in country 2 than in country 1, the level of migration the social planner of country 2 implements is less than the one chosen by the social planner of country 1.

2.
There are always incentives for illegal migration from country 1 toward country 2.
Proof.
To find the optimal level of migrants, we equalize \(\hat K^{i}\left (m^{i}\right)={K_{w}^{i}}\) so that \(m^{i \star }= \Psi ^{1}\left ({K_{w}^{i}}\right).\) This leads to the expression of the welfare maximizing level of migrants for each country
$$\begin{aligned} {m^{2}}^{\star}&=\left[\left[\frac{1\sigma+\beta(1\sigma\nu)}{\beta\sigma\left(1\varepsilon^{2}\right)}\right] \frac{{\ell^{2}}^{\sigma}_{w}}{{\ell^{2}}^{\sigma}}\frac{{\theta^{2}}^{\nu}_{w}}{{\theta^{2}}^{\nu}}\right]^{\frac{1}{\nu+\sigma\left(1\varepsilon^{2}\right)}}1, \\ {m^{1}}^{\star}&=1\left[\left[\frac{1\sigma+\beta(1\sigma\nu)}{\beta\sigma\left(1\varepsilon^{1}\right)}\right] \frac{{\ell^{1}}^{\sigma}_{w}}{{\ell^{1}}^{\sigma}}\frac{{\theta^{1}}^{\nu}_{w}}{{\theta^{1}}^{\nu}}\right]^{\frac{1}{\nu+\sigma\left(1\varepsilon^{1}\right)}}. \end{aligned} $$
□
Lemma 3.
Since \({\ell ^{i}}_{w}, \ell ^{i}, {\theta ^{i}_{w}}\) and \(1/\left [\left (1\varepsilon ^{i}\right){\theta ^{i}}^{\nu }\right ]\) are increasing functions of the return to education ε
^{i},i=1,2,m
^{2}
^{⋆} is an increasing convex function of ε
^{2}, and m
^{1}
^{⋆} is a decreasing concave function of ε
^{1}.
The proof is given in Appendix Appendix B.
Proposition 3.
There are incentives for illegal migrations.
Proof.
Incentive for migration are directed from country 1 to country 2, if and only if ∣m
^{2}
^{⋆}∣<∣m
^{1}
^{⋆}∣, and in the remaining of the paper, we will assume that this condition holds. If not, international migration is in the opposite direction. □
In what follows, since we study postmigration perfect foresight equilibria, the postmigration flow is defined \(m=\min \left \{ m^{1}, m^{2} \right \}\), which is exactly anticipated by each country.
5.4 Incentive for illegal migration
Each social planner maximizes the utility of his own country; consequently, all education, consumption, labor, and capital are set at their welfare maximizing levels. When borders are open, there exists m
^{i}
^{⋆} so that \(\hat K^{i}\left ({m^{i}}^{\star }\right)={K_{w}^{i}}\) is satisfied.
Proposition 4.
In postmigration steadystate equilibrium, there are incentives for illegal migration.
Proof.
Let us consider the case were the optimal desired flows of migrants differ across countries, since ε
^{2}<ε
^{1}. In that case and under the unilateral migration condition, the two migration flows satisfy the following inequality ∣m
^{1}
^{⋆}∣≥∣m
^{2}
^{⋆}∣. Consequently, there are incentives for country 1 to support illegal migration flows in the direction of country 2. □
5.5 The emergence of an optimal price differential between countries
Proposition 5.
In postmigration steadystate equilibrium, there is no price equalization across countries.
Proof.
Since the returns to education differ across countries, the optimal migration policies lead the economies to different steadystate equilibria. Indeed, we have two main cases

1.
The first case is such that m
^{1}
^{⋆}≥m
^{2}
^{⋆}, so that country 2 reaches the optimal level “before” country 1. In such a case, \(\hat {K}^{1}\left ({m^{2}}^{\star }\right)<{K_{w}^{1}}\) and \(\hat {K}^{2}\left ({m^{2}}^{\star }\right)={K_{w}^{2}}\). Consequently, by assumption on the returns to education, ε
^{1}>ε
^{2}, we necessarily have \(\hat {K}^{1}\left ({m^{2}}^{\star }\right)<\hat {K}^{2}\left ({m^{2}}^{\star }\right)\).

2.
The second case is such that m
^{1}
^{⋆}<m
^{2}
^{⋆}, so that country 1 reaches the optimal level “before” country 2. In such a case, \(\hat {K}^{1}\left ({m^{1}}^{\star }\right)={K_{w}^{1}}\) and \(\hat {K}^{2}\left ({m^{1}}^{\star }\right)>{K_{w}^{2}}\) according to our assumptions. Consequently, \(\hat {K}^{1}\left ({m^{1}}^{\star }\right)>\hat {K}^{2}\left ({m^{1}}^{\star }\right)\).
A natural consequence of such differences in steadystate capital is that there is no price equalization across countries. It always remains a wage differential \(\overline w^{1}\neq \overline w^{2}\). \(\overline p^{1}\neq \overline p^{2}\), and there is an interest rate differential across countries, \(\overline R^{1}\neq \overline R^{2}\). □
Moreover, rewritting (19) and (20) to obtain the steady–state capital per individual leads us to conclude that the wage of the sending country increases with migrants, and the interest rate decreases with migrants. Such a result is compatible with the implicit legal system of migration, which sets high wages and low interest rates in order to refrain individuals from migrating.